Optimal Control Operator Perspective and a Neural Adaptive Spectral Method

TL;DR

This work reframes optimal control as learning an instance-to-solution operator $\mathcal{G}:I\to U$ that maps OCP instances to their optimal controls, enabling one-shot inference without explicit dynamics or iterative optimization. It introduces Neural Control Operator (NCO) realized by Neural Adaptive Spectral Method (NASM), an adaptive-basis neural operator that computes $\hat{\vb{u}}(t)$ from problem instances using time- and instance-dependent coefficients and an aggregation mechanism, with a formal approximation-error bound. Empirically, NASM achieves massive inference-time speedups (often $>10^3$–$10^4$×) and strong ID/OOD generalization across synthetic and real OCPs, including planary pushing and quadrotor control, outperforming several neural-operator baselines. The results indicate that NASM’s architectural design—adaptive basis, time-conditioned coefficients, and operator-learning—provides robust, reusable control solutions suitable for high-dimensional, data-rich control tasks, with clear avenues for enforcing constraints via physics-informed extensions.

Abstract

Optimal control problems (OCPs) involve finding a control function for a dynamical system such that a cost functional is optimized. It is central to physical systems in both academia and industry. In this paper, we propose a novel instance-solution control operator perspective, which solves OCPs in a one-shot manner without direct dependence on the explicit expression of dynamics or iterative optimization processes. The control operator is implemented by a new neural operator architecture named Neural Adaptive Spectral Method (NASM), a generalization of classical spectral methods. We theoretically validate the perspective and architecture by presenting the approximation error bounds of NASM for the control operator. Experiments on synthetic environments and a real-world dataset verify the effectiveness and efficiency of our approach, including substantial speedup in running time, and high-quality in- and out-of-distribution generalization.

Figures (5)

• Figure 1: Phase-2 cost curves of two failed instances of two-phase control hwang2022solving on Pendulum system. The control function gradually moves outside the training distribution of phase 1. As a result, the control function converges w.r.t. the cost predicted by the surrogate model (blue), but diverges w.r.t. true cost (red).
• Figure 2: The architecture of NASM. The network takes two inputs: OCP instance $i$ and time index $t$. The input $i$ is pre-processed by the Encoder. Then both $t$ and encoding $\vb*{e}$ are fed into the Coefficient Network to obtain coefficients $\vb*{c}$ and adaptive parameters $\vb*{\theta}$. The adaptive basis (e.g. Fourier series) outputs function values $\vb*{b}$, which is multiplied with $\vb*{c}$ and aggregated to the final output $\hat{\vb*{u}}(t)$, the estimation of optimal control for instance $i$ at time $t$. Detailed explanation is given in Section \ref{['sec:arch']}.
• Figure 3: Inference time and mean absolute percentage error (MAPE) on in-distribution (ID) and OOD benchmarks. NASM (red bars) achieves higher or comparable accuracy, with the fastest or second fastest speed.
• Figure 4: Pushing environmentyu2016more.
• Figure 5: List of variables explored in Pushing dataset, credited to yu2016more.

Theorems & Definitions (10)

• Theorem 1: Approximator Error, Informal
• Theorem 2: Decomposition of NASM Approximation Error
• Theorem 3: Fourier Reconstructor Error
• Theorem 4: Approximator Error (MLP backbone)
• Theorem 5: DeepONet Reconstructor Error lanthaler2022error
• Lemma 6: Approximation Error of one-dimensional MLP guhring2020error
• Lemma 7: Approximation Error of $\mathbf{p}$-dimensional MLP
• Lemma 8: Equivalence of approximation between MLPs and CNNs petersen2020equivalence
• Lemma 9: Approximation Error of CNN
• proof